arXiv research

Contact Lie systems analyze integral curves of Hamiltonian vector fields.

New Lie systems defined on $k$-contact manifolds, with applications.

Study new $k$-contact distributions and Lie systems.

New method for constructing contact Lie systems on various spaces.

Reduces equations for contact mechanical systems on Lie groups by exploiting symmetries.

Develops integrators for contact Hamiltonian systems preserving geometric structure.

2-compatible Lie algebras are quadratic deformations of Lie algebras with specific constraints.

Develops k-contact geometry theory for field theories.

Lie contact structures generalize the classical Lie sphere geometry of oriented hyperspheres in the standard sphere. They can be equivalently described as parabolic geometries corresponding to the contact grading of orthogonal real Lie algebra. It follows the underlying geometric structure can be interpreted in several…

Paper bridges quantum and classical mechanics for open systems.

Proofs centerless unimodular contact Lie algebras.

New systems of linear PDEs discovered in 3D contact manifolds.

A result from Gromov ensures the existence of a contact structure on any connected non-compact odd dimensional Lie group. But in general such structures are not invariant under left translations of the Lie group. The problem of finding which Lie groups admit a left invariant contact structure (contact Lie groups), is t…

Prove an isoperimetric inequality for compact bodies in 3D contact non-unimodular Lie groups.

We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive some obstruction results to the existence of left invariant contact structu…

We prove that a K-contact Lie group of dimension five or greater is the central extension of a symplectic Lie group by complexifying the Lie algebra and applying a result from complex contact geometry, namely, that, if the adjoint action of the complex Reeb vector field on a complex contact Lie algebra is diagonalizabl…

The paper classifies structures on specific Lie groups.

Study of curves in Lie sphere geometry using moving frames and variational principles.

Study proves all left-invariant contact structures on 3D Lie groups are tight.

Contact Lie algebras have specific properties related to stabilizers and invariant polynomials.

We study Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds. The almost-cosymplectic-contact structure admits on the sheaf of pairs of 1-forms and functions the structure of a Lie algebra. We describe Lie subalgebras in this Lie algebra given by…

We study the question of the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group. We find a complete list of the 22 classes of seven-dimensional n…

The paper finds a contact form on SL(2p) for p > 1.

This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., compact manifolds that are the quotient of some Lie group G with a left invariant contact structure and a uniform lattice subgroup. We re-examine Alexander's criteria for existence of lattices on solvable…

In this paper, the symmetry group of a differential system of n quadratic homogeneous first order ODEs of n variables is studied. For this purpose, we consider the action of both point and contact transformations to signify the corresponding Lie algebras. We also find the independent differential invariants of these ac…

Classifies semisimple symmetric contact spaces under Lie groups.

Two reduction schemes for symplectic manifolds are shown equivalent.

Study on a specific type of Lie algebras with Kähler and contact properties.

New contact structures defined on differentiable stacks.

In this paper we generalize the main notions from the geometry of (almost) contact manifolds in the category of Lie algebroids. Also, using the framework of generalized geometry, we obtain an (almost) contact Riemannian Lie algebroid structure on a vertical Liouville distribution over the big-tangent manifold of a Riem…

In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Plücker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Plücker…

The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some kno…

The paper is devoted to the complete classification of all real Lie algebras of contact vector fields on the first jet space of one-dimensional submanifolds in the plane. This completes Sophus Lie's classification of all possible Lie algebras of contact symmetries for ordinary differential equations. As a main tool we …

We study left invariant contact forms and left invariant symplectic forms on Lie groups. We give the classification of all symplectic structures on nilpotent Lie algebras up the dimension 6.

The paper classifies para-Kähler structures on Lie groups.

The paper develops methods to derive mixed superposition rules for Lie systems and applies them to various physical systems.

Classifies contact seaweeds based on their algebraic properties.

We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. W…

In this paper we study some affine structures on nilpotent Lie algebras endowed with a contact form. These affine structures are constructed from an affine structure on a symplectic Lie algebra by a central extension.

Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.

Smooth contact mappings in a flat (2,3,5)-distribution are shown to be smoother.

We consider the Lie algebra of all vector fields on a contact manifold as a module over the Lie subalgebra of contact vector fields. This module is split into a direct sum of two submodules: the contact algebra itself and the space of tangent vector fields. We study the geometric nature of these two modules.

This thesis extends contact structures to differentiable stacks using line bundle-valued 1-forms.

Characterizes G2-structures on Lie algebras with non-trivial center.

We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensiona…

We study the indefinite metric $G$ in the contact phase space $(P,θ)$ of a homogeneous thermodynamical system introduced by R. Mrugala. We calculate the curvature tensor, Killing vector fields, second fundamental form of Legendre submanifolds of $P$ - constitutive surfaces of different homogeneous thermodynamical syste…

In this paper, we introduce generalized almost para-contact manifolds and obtain normality conditions in terms of classical tensor fields. We show that such manifolds naturally carry certain Lie bialgebroid/quasi-Lie algebroid structures on them and we relate this new generalized manifolds with classical almost para-co…

Almost contact B-metric manifolds of dimension 3 are constructed by a two-parametric family of Lie groups. The class of these manifolds in a known classification of almost contact B-metric manifolds is determined as the direct sum of the main vertical classes. The type of the corresponding Lie algebras in the Bianchi c…